Question

Find the coordinates of the foot of the perpendicular drawn from the point $A(1,2,1)$ to the line joining $B(1,4,6)$ and $C(5,4,4)$

• A. $(2,3,6)$
• B. $(1,4,4)$
• C. $5,4,6)$
• D. $(3,4,5)$

Answer 1

The correct option is D $(3,4,5)$

The equation of line through $B(1,4,6)$ and $C(5,4,4)$ is

$\frac{x-1}{5-1}=\frac{y-4}{4-4}=\frac{z-6}{4-6}$

or $\frac{x-1}{4}=\frac{y-4}{0}=\frac{z-6}{-2}=k$

or $x-1=4k,y-4=0,z-6=-2k$

or $x=4k+1,y=4,z=-2k+6$

Any point $D$ on it is $(4k+1,4,-2k+6)$

Let $D$ be the foot of the perpendicular from $A$ on $BC$

Direction ratios of $AD$ are $4k+1-1,4-2,-2k+6-1$

Direction ratios of $AD$ are $4k,2,-2k+5$

Direction ratios of $BC$ are $4,0,-2$

Since $AD$ is perpendicular to $BC$

$\therefore 4k\times 4+2\times 0+(-2k+5)(-2)=0$

$\therefore 16k+0+4k-10=0$

$\therefore 20k=10$ or $k=\frac{1}{2}$

$\therefore D$ is $(2+1,4,-1+6)$ or $(3,4,5)$